If and are the roots of x^2-2x+2=0, then the least value of n for which ()^n=1 is:

Question

If $\alpha$ and $\beta$ are the roots of $x^2-2x+2=0$, then the least value of $n$ for which $\left(\dfrac\alpha\beta\right)^n=1$ is:

Accepted Answer

$\alpha,\beta=1\pm i$. $\dfrac\alpha\beta=\dfrac{1+i}{1-i}=i$ (or $-i$).

$i^n=1 \implies n$ a multiple of 4. Least such $n$ is 4.

Solutions · 1